Polynomial stabilization of some dissipative hyperbolic systems

TL;DR

This paper investigates the stabilization of dissipative hyperbolic systems, specifically acoustic systems with spatial damping, establishing conditions for exponential and polynomial decay of solutions over time.

Contribution

It introduces new conditions under which hyperbolic systems exhibit exponential or polynomial decay, using time domain and frequency domain methods respectively.

Findings

Exponential decay achieved via a time domain approach.

Polynomial decay established through a frequency domain method.

Conditions identified for different decay rates based on damping properties.

Abstract

We study the problem of stabilization for the acoustic system with a spatially distributed damping. Imposing various hypotheses on the structural properties of the damping term, we identify either exponential or polynomial decay of solutions with growing time. Expo- nential decay rate is shown by means of a time domain approach, reducing the problem to an observability inequality to be verified for solutions of the associated conservative problem. In addition, we show a polynomial stabilization result, where the proof uses a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.